The name is a reference to the 1415 battle Agincourt/Azincourt opposing French and English armies, therefore it's 1. c4 e6 (French vs English once again).
Agincourt Defense

this is a fallacy i sometimes call " i play the king's gambit vs the sicilian" fallacy a lot of beginners make in categorizing openings.
To answer your question in a brute way, yes, the agincourt is a french-like formation reply to the english, but thats why by definition its not a french! only system openings and system defenses retain their name that way regardless of what the opponent plays.
it should be obvious that despite the pieces going in roughly the same places the french defense and the agincourt play completely different because of whites formation of choice. (the agincourt can sometimes transpose to queen gambit declined if white plays an early d4, and these also have a different flavor although maybe more similar than the comparison to the french)
Is this just the French Defense, but when the opponent starts off with the English instead of 1. e4?